From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5762 Path: news.gmane.org!not-for-mail From: John Baez Newsgroups: gmane.science.mathematics.categories Subject: RE : bilax monoidal functors Date: Fri, 7 May 2010 20:27:18 -0700 Message-ID: Reply-To: John Baez NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1273359418 23172 80.91.229.12 (8 May 2010 22:56:58 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Sat, 8 May 2010 22:56:58 +0000 (UTC) To: categories Original-X-From: categories@mta.ca Sun May 09 00:56:56 2010 connect(): No such file or directory Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1OAsx1-0003qj-35 for gsmc-categories@m.gmane.org; Sun, 09 May 2010 00:56:51 +0200 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1OAsS6-0001ay-ER for categories-list@mta.ca; Sat, 08 May 2010 19:24:54 -0300 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5762 Archived-At: Andr=C3=A9 Joyal wrote: I am using the following terminology for > higher braided monoidal (higher) categories: > > Monoidal< braided < 2-braided <....... > A (n+1)-braided n-category is symmetric > according to your stabilisation hypothesis. > > Is this a good terminology? > I use "k-tuply monoidal" to mean what you'd call "(k-1)-braided". This seems preferable to me, not because it sounds nicer - it doesn't - but because it starts counting at a somewhat more natural place. I believe tha= t counting monoidal structures is more natural than counting braidings. For example, a doubly monoidal n-category, one with two compatible monoidal structures, is a braided monoidal n-category. I believe this is a theore= m proved by you and Ross when n =3D 1. This way of thinking clarifies the relation between braided monoidal categories and double loop spaces. Various numbers become more complicated when one counts braidings rather than monoidal structures: An n-tuply monoidal k-category is (conjecturally) a special sort of (n+k)-category... while an n-braided category is a special sort of (n+k+1)-category. Similarly: n-dimensional surfaces in (n+k)-dimensional space are n-morphism= s in a k-tuply monoidal n-category... but they are n-morphisms in an (k-1)-braided n-category. And so on. On the other hand, if it's braidings that you really want to count, rather than monoidal structures, your terminology is perfect. By the way: I don't remember anyone on this mailing list ever asking if their own terminology is good. I only remember them complaining about othe= r people's terminology. I applaud your departure from this unpleasant tradition! Best, jb [For admin and other information see: http://www.mta.ca/~cat-dist/ ]